Absolute rigging fails as my mesh is distorted
Well from the second image your issue is that not all meshes are parented to the bones.

First off, use the mesh with already applied weight painting. Go to weight painting option when selection the mesh and then see where the weights of bones are applied under Vertex groups (downward triangle).

Go through each bone and if you added automatic weights, edit them by selecting Draw or delete extras with Subtract brush on top left.